where \(B_r^{\mathbb {H}^2}(\bar{x})\)  is the ball of radius \(r\)  and centre \(\bar{x}\)  in the hyperbolic plane. Moreover it is known (see for instance the top of [1]}) that
\(\lambda _1^D(B_r^{\mathbb {H}^2}(\bar{x}))\le \frac{1}{4}+\left(\frac{2\pi }{r}\right)^2,\) 
